Abstract
Given a module M over a commutative unital ring R, let denote the covering number, i.e. the smallest (cardinal) number of proper submodules whose union covers M; this includes the covering numbers of Abelian groups, which are extensively studied in the literature. Recently, Khare–Tikaradze [Comm. Algebra, in press] showed in several cases that
where SM is the set of maximal ideals
with
Our first main result extends this equality to all R-modules with small Jacobson radical and finite dual Goldie dimension. We next introduce and study a topological counterpart for finitely generated R-modules M over rings R, whose ‘some’ residue fields are infinite, which we call the Zariski covering number
To do so, we first define the “induced Zariski topology” τ on M, and now define
to be the smallest (cardinal) number of proper τ-closed subsets of M whose union covers M. We then show our next main result:
for all finitely generated R-modules M for which (a) the dual Goldie dimension is finite, and (b)
whenever
is finite. As a corollary, this alternately recovers the aforementioned formula for the covering number
of the aforementioned finitely generated modules. Finally, we discuss the notion of κ-Baire spaces, and show that the inequalities
again become equalities when the image of M under the continuous map
(with appropriate Zariski-type topologies) is a κM-Baire subspace of the product space.
2020 MATHEMATICS SUBJECT CLASSIFICATION:
Acknowledgments
The author is thankful to Prof. Amartya Kumar Dutta, Indian Statistical Institute, Kolkata for introducing him to the toy problem for vector spaces, and for motivating him to explore this in a broader setting and in greater depth. Further, the author is extremely grateful to Prof. Apoorva Khare, Indian Institute of Science, Bangalore, for several stimulating and motivating interactions, as well as for a careful and detailed reading of previous versions of this manuscript, which helped greatly improve the exposition.